Delta-8-tetrahydrocannabinol (Δ8-THC) provides similar medicinal effects of the well-known psychoactive constituent of Cannabis genus delta-9-tetrahydrocannabinol (Δ9-THC). By interacting with the endocannabinoid system (ECS), these cannabinoids mediate the same therapeutic response acting as antiemetic, anxiolytic, appetite stimulant, analgesic and neuroprotective agents.  The strong aspect making Δ8-THC so interesting is that it has half potency  of Δ9-THC in terms of psychoactivity and doesn’t induce paranoia as sometimes the latter does.
The anti-anxiety qualities of Δ8-THC have been related to its ability to interact differently with the ECS than its Δ9 counterpart: the particular structure of Δ8-THC makes the binding with CB1 and CB2 receptors less effective leading to milder psychotropic consequences.
The principal role of ECS is maintaining the homeostasis of the human body, working as a cell-signalling system and regulating mood, sleep, pain, learning memory, appetite, digestion, etc.
Within other functions, ECS is involved in regulating negative emotions through CB1 receptors: when blocked by an antagonist or by a full agonist, they elicit signals promoting anxiety and fear to alert the organism to potentially dangerous or harmful stimuli. 
Being a partial agonist of CB1 receptors, Δ8-THC is capable of activating a response to stress and reducing anxiety.  Even if Δ9-THC is a partial agonist too, the paranoia caused in some subjects is linked to its high binding affinity to CB1 receptors, meaning that small molecular structure differences can lead to big discrepancies in biological effects.
The important aspect to remember is that the anxiolytic efficacy of Δ8-THC is dependent on the dosage: a low dose of Δ8-THC could be useful to relief stress, whilst an excess of it could heighten anxiety symptoms. Moreover it is necessary to be aware of the exact composition and purity of Δ8-THC products: if the desired benefit is the anti-anxiety aspect, it is better to look for Δ9-THC-free formulations.
References: https://pubchem.ncbi.nlm.nih.gov/compound/2977#section=Depositor-Supplied-Patent-Identifiers  Hollister L.E. et al. Delta-8- and delta-9-tetrahydrocannabinol; Comparison in man by oral and intravenous administration. Clinical Pharmacology & Therapeutics, (1973). 14(3), 353–357. doi:10.1002/cpt1973143353 [Journal impact factor = 7.266 ] [Times cited = 36]  Ruehle, S. et al. The endocannabinoid system in anxiety, fear memory and habituation. Journal of Psychopharmacology, (2012). 26(1), 23–39. doi:10.1177/0269881111408958 [Journal impact factor = 7.73 ] [Times cited = 206]  Walter L. et al. Cannabinoids and neuroinflammation. British Journal of Pharmacology, (2004) 141(5), 775–785.doi:10.1038/sj.bjp.0705667 [Journal impact factor = 4.738 ] [Times cited = 343]
Image: Medical grade cannabis_Picture Sabina Pulone